Considerable evidence suggests that during the progression of complex diseases, the deteriorations are not necessarily smooth but are abrupt and may cause a critical transition from one state to another at a tipping point. Here, we develop a model-free method to detect early-warning signals of such critical transitions, even with only a small number of samples. Specifically, we theoretically derive an index based on a dynamical network biomarker (DNB) that serves as a general early-warning signal indicating an imminent bifurcation or sudden deterioration before the critical transition occurs. Based on theoretical analyses, we show that predicting a sudden transition from small samples is achievable provided that there are a large number of measurements for each sample, e.g., high-throughput data. We employ microarray data of three diseases to demonstrate the effectiveness of our method. The relevance of DNBs with the diseases was also validated by related experimental data and functional analysis.
It has been identified that a sudden change of a system state exists widely in ecosystems1,2, climate systems3,4, economics and global finance5,6. Such a change often occurs at a critical threshold, or the so-called “tipping point”, at which the system shifts abruptly from one state to another. This is well known in dynamical systems theory as a bifurcation that results in a qualitative transition in states or attractors7,8. Recently, evidence has been found suggesting that the similar phenomena exist in clinical medicine, that is, during the progression of many complex diseases, e.g., in chronical diseases such as cancer, the deterioration is not necessarily smooth but abrupt9,10,11,12,13. In other words, there exists a sudden catastrophic shift during the process of gradual health deterioration that results in a drastic transition to a disease state. In order to describe the underlying dynamical mechanism of complex diseases, their evolutions are often modeled as time-dependent nonlinear dynamical systems, in which the abrupt deterioration is viewed as the phase transition at a bifurcation point, e.g., for prostate cancer14, asthma attacks9 and epileptic seizures15. We are particularly interested in the complex diseases with sudden deterioration phases or critical transition points during their progressions. Depending on the progression level of such illnesses, we divide the process into three stages: i.e., a normal state, a pre-disease state (or a critical state) and a disease state (Figs. 1 a and b–d). The normal state is a steady state, representing a relatively healthy stage during which the disease is under control, in an incubation period or in a chronic inflammation period. The pre-disease state has been discussed in many previous works (e.g., Achiron et al.16). It is usually defined as the limit of the normal state immediately before the tipping point is reached. In this pre-disease stage (see Figs. 1 a and c), the process is usually reversible to the normal state if appropriately treated, implying instability of the pre-disease state. However, it usually becomes irreversible to the normal state if the system passes the critical point and enters another stable state, or the unique attractor at the disease stage (Figs. 1a and d). Hence, it is crucial to detect the pre-disease state so as to prevent qualitative deterioration by taking appropriate intervention actions.
Recently, topics on predicting the tipping point are increasingly receiving much attention in many scientific fields17, e.g., in ecological systems1,18,19,20, climate systems3,21,22 and physiological systems10. Many such predictions are mainly based on “critical slowing down”23, a generic dynamical phenomenon occurring in the vicinity of a bifurcation point where the system becomes increasingly slow when recovering from small perturbations back to its equilibrium state24,25. Considerable efforts have been made in successfully finding general early-warning signals, ranging from theoretical derivation to real data analysis, which have shown that the increases in variance24, autocorrelation21, skewness26 and spatial correlation27,28 are generally expected to occur as a system approaches to a critical point.
However, in the case of complex diseases, it is notably hard to predict such critical transitions for the following reasons. First, because a pre-disease state is a limit of the normal state, the state of the system may show little apparent change before the tipping point is reached. Thus, the diagnosis by traditional biomarkers and snapshot static measurements may not be effective to distinguish those two states (Figs. 1b, c). Second, despite considerable research efforts, no reliable disease model has been developed to accurately detect the early-warning signals. In particular, deterioration processes may be considerably different even for the same subtype of a disease, depending on individual variations, which makes model-based prediction methods fail for many cases. Third and most importantly, detecting the pre-disease state must be an individual-based prediction, however, usually there are only a few of samples available for each individual, unlike many other complex systems that are measured over a long term with a large number of samples.
On the other hand, rapid advancements on high-throughput technologies have enabled us to observe gene expressions or even protein expressions at the genome-wide scale29, i.e., with over thousands of measurements in one sample. Such high-dimensional data not only provides a global view with rich information on the concerned system, but also represents the accumulated effects of its long-term dynamics30, which implies that even a small number of samples may characterize the dynamical features of a living organism. In this work, by using time-course high-throughput omic data and based on the principle of critical slowing down or a bifurcation process, we proposed a novel model-free (network-based) method to detect early-warning signals of complex diseases even with a small number of samples. Specifically, we theoretically showed that a dynamical network biomarker (DNB) can serve as a general early-warning signal indicating an imminent bifurcation or sudden deterioration before the critical transition occurs. We successfully identified critical transitions as well as DNBs for lung injury disease, liver cancer and lymphoma cancer based on their microarray data. The relevance of the identified DNBs with the diseases was also validated by pathway enrichment analysis, functional analysis, bootstrap analysis and related experimental data. It should be note that we aimed to detect the pre-disease state by finding its associated DNB, rather than by distinguishing between the disease state (e.g., disease samples) and the normal state (e.g., control samples) by finding its biomarkers.
Criteria for detecting early-warning signal
In particular, we theoretically derived network-based dynamical criteria that serve as a general early-warning signal indicating the imminent bifurcation before a critical transition occurs. Based on theoretical analysis, we proved that the following generic properties hold when the system reaches the pre-disease state (see the derivation of DNBs in Methods, Figs. 1e–h and Supplementary Information A for details).
There exists a group of molecules, i.e., genes or proteins, whose average Pearson's correlation coefficients (PCCs) of molecules drastically increase in absolute value.
The average PCCs of molecules between this group and any others (i.e., between molecules inside this group and any other molecules outside this group) drastically decrease in absolute value.
The average standard deviations (SDs) of molecules in this group drastically increase.
If all of these three conditions are satisfied simultaneously, we call this group a dominant group of the system, whose change will reflect a transition of the system to the disease state. Actually, each of the three conditions represents a criterion and their combination is naturally expected to be a strong signal or an indicator for the pre-disease state. Because the dominant group characterizes dynamical features of the underlying system and the molecules in the group are also strongly and dynamically correlated in the pre-disease state, the molecules in the group are expected to form a subnetwork from a network viewpoint. Hence, we regard it as a dynamical network of biomarkers, or a dynamical network biomarker (DNB). Unlike the traditional molecular biomarkers used in medicine, whose expressions reflect the presence or severity of the disease state and are required to have consistent (or constant) values that are different in the respective disease and normal states, the DNB is a strongly correlated molecular subnetwork where the concentrations of molecules, however, dynamically change without keeping constant values in the pre-disease state as shown in Fig. 1h. In other words, the concentrations of molecules in the DNB tend to increasingly fluctuate when the system approaches to the pre-disease state, although they behave dynamically in a strongly collective manner, which is a key feature of the DNB. This is why it can be used to detect the early signal of a complex disease in the early stage, which is not otherwise possible using traditional biomarkers or methods. Hence, the existence of the DNB implies that the system is in the pre-disease state. Note that each individual may have a different DNB even for the same disease due to individual variations. Therefore, in contrast to traditional biomarkers, a DNB is not necessarily composed of a fixed bunch of molecules even for the same disease but may have different members depending on individual features that are detected from individual high-throughput data.
Composite index for pre-disease state
We then describe the theoretical background of the DNB for model-free prediction and we derive a general quantitative index as an early-warning signal for the pre-disease state regardless of the detailed differences among systems.
Assume that the progression of a disease can be expressed by the following dynamical system
Z(k) = (z1(k), …, zn(k)) represent observed data, i.e., concentrations of molecules (e.g., gene expressions or protein expressions) at time k (k = 0, 1, …), e.g., hours or days, which are the variables describing the dynamical state of the system. P are parameters representing slowly changing factors, including genetic factors (e.g., SNP and CNV) and epigenetic factors (e.g., methylation and acetylation), which drive the system from one state (or attractor) to another (see Supplementary Fig. S1). f = (f1, …, fn) are generally nonlinear functions of Z(k) with a fixed point , such that . Assume that there is a value Pc such that at least one of the eigenvalues of the Jacobian matrix equals 1 in modulus, which implies that the system undergoes a bifurcation at when P reaches the threshold Pc. By analyzing the nonlinear dynamics near the bifurcation point, we can theoretically derive the above three criteria to detect the dominant group or DNB (see Methods and Supplementary Information A). To obtain a strong signal in the pre-disease state, we further combine the three criteria together to construct a composite index I as follows:
where PCCd is the average PCC of the dominant group in absolute value; PCCo is the average PCC between the dominant group and others in absolute value; and SDd is the average SD of the dominant group.
Although the expression of each zi may stochastically change at any time instant due to the fluctuation, the composite index is expected to increase sharply whenever the system approaches to a critical transition point and therefore, it can serve as an effective early-warning signal to identify the pre-disease state.
According to the derivation, the warning signal is generated by the interdependent molecules in the dominant group, through their dynamical interactions near the critical transition. Because the molecules in the dominant group are strongly and dynamically correlated, they behave as a dynamical subnetwork or a module in the pre-disease state, forming the DNB, which will move to the disease state first when the system undergoes further perturbations of parameters into the disease stage. These properties should be generic and lie in many complex diseases with sudden deterioration phenomena. Clearly, analysis based on such a signal is applicable to diseases with or without a model. The algorithm to numerically identify the DNB from high-throughput data is given in detail in Supplementary Information C. Note that the purpose of this work is not to identify the parameters or factors that first drive the system into the disease state, but to identify the responsive DNB which first moves into the disease state driven by any known or unknown factors (or first reflects such a critical transition). Some molecules related to the driving factors (e.g., SNPs or CNVs) may or may not be in the DNB.
We simulated an example of a five-node gene regulatory network (see Fig. 2a) illustratively to demonstrate the DNB and the composite index. The detailed descriptions of the network represented by a set of stochastic differential equations are provided in Supplementary Information B and numerical simulations are shown in Fig. 2, which demonstrates the predictive power of the DNB.
Application to complex diseases
We further conducted the prediction for three real diseases using high-throughput experimental data, which are respectively the microarray data of a lung injury with carbonyl chloride inhalation exposure (or acute lung injury)31, the hepatic lesion by chronic hepatitis B (or HBV induced liver cancer)32 and the B-cell lymphomagenesis of transgenic murine (or lymphoma)33. The detailed algorithm and data description are described in Supplementary Information C and D, respectively. For the diseases of phosgene inhalation lung injury and hepatic lesion, Fig. 3 shows that the composite indices based on DNBs during the disease progression are consistent with the observed biological phenotypes. The analysis for the B-cell lymphomagenesis is presented in Supplementary Information D. To explore the biological implication, functional analysis was implemented on these identified DNBs from the gene expression datasets respectively. Figures 4a–d show the DNBs with the functional interactions (protein-protein interactions and TF-target regulations) for the diseases. In particular, we take the acute lung injury as a concrete example to present the calculation procedure in detail (see Supplementary Information D). The dynamical change in the network structure and expression variations for the identified DNB are shown in Figs. 4e–h. Clearly, the DNB forms a strongly correlated subnetwork to provide the significant warning signal near the critical state (8 h). The dynamical changes in the DNBs over whole periods are shown in Supplementary Fig. S8.
To further compare the DNB dynamics with other molecules, we also graphically demonstrate the dynamics of the whole mouse molecular network (protein-protein interactions and TF-target regulations) including the DNB in Figs. 4i–l (also see the whole mouse network for other periods in Supplementary Fig. S9), which clearly shows the significance of the DNB in terms of expression variations and network structures near the critical state (8 h). In particular, before the disease state, there are no significant differences between the members of the DNB and other genes during all periods except at 8 h, when the members of the DNB behave in a considerably different manner in terms of their expression variations and network connections by satisfying our three criteria. However, after the system is driven into the disease state, interestingly the members of the DNB appear to behave in a manner similar to other genes again (e.g., Fig. 4l).
In order to verify the biological significance of the identified DNBs, we carried out bootstrap analysis respectively for the three diseases, as described in Supplementary Information E. For the phosgene inhalation lung injury (or acute lung injury) and hepatic lesion (or HBV induced liver cancer), some enriched GO functions and dysfunctional pathways underlying the DNBs are listed in Table 1. Part of the identified DNBs are also shown in Table 1 (see Supplementary Table ‘Identified DNBs’ for complete lists). The details and full lists are presented in Supplementary Information F.
The functional analysis of these cases shows that the DNBs are closely relevant to the corresponding complex diseases, thereby validating the effectiveness and advantage of our method. In the lung injury study, we found that the dysfunctions of the DNB are consistent with the corresponding mechanisms of the phosgene-induced injury (or acute lung injury), especially for antioxidant reactions, dysfunctional metabolic processes, inflammatory regulation and regulations of cell proliferation and death31. From pathway enrichments, we found that pathways, such as pathways of repairing DNA damage (e.g., p53 signaling pathways), pathways of inflammatory (e.g., acute myeloid leukemia) and pathways of reducing oxidation (e.g., glutathione metabolism), are highly related to the dysfunctional mechanisms of the aggressive disease31. In particular, some genes with the most significant changes are related to the antioxidant reflections and the response of reducing DNA damage. Some of them are involved in the related glutamine-metabolic activities (e.g., ASNS and GCLC) and participate in oxidation reduction (e.g., SRXN1, PGD, TXNL1, HMOX1, CYP51 and CH25H). Moreover, owing to protection against the denaturation of proteins and lipoids caused by acylation of phosgene and tissue damage, heat shock protein activities (e.g., HSPA1A, HSPA1b, DNAJC5, DNAJB4) and regulation of apoptosis (e.g., PMAIP1, GADD45G, EPHA2) were identified in the DNB. Furthermore, 43 of the 262 genes and TFs (hypergeometric test, p-value < 7.89e-36) were validated with significantly close relation with lung injury by phosgene31 (Supplementary Table ‘KEGG enrichment analysis’).
In HBV infection and progression, the dysfunction of the regulation system causes the hepatitis B disease, which may lead to cirrhosis and hepatocellular carcinoma32. From the KEGG pathway enrichment results, the DNB members are included in many HBV-related pathways, such as pathways in cancer, hepatitis C, inflammatory activation (e.g., acute myeloid leukemia) and DNA repair (e.g., cytosolic DNA sensing pathway). The enriched GO functions underlying the DNB are particularly related to immune systems that are activated to protect against HBV and inordinate dysfunctions associated with the performance in the viral life cycle. In the DNB, HLA-DMA and B2M, which play crucial roles in antigen presentation and STAT6, which is associated with the switch of antibody isotypes, undergo significant changes during the disease development32. Further, some genes in the basic cellular processes were expressed in a disorderly manner, e.g., HDAC10 and NAP1L1, which are associated with the regulation of cell death and abnormal reaction in transcription and translation and GCC2 and CDH1, which are associated with transportation and localization. Moreover, CDH1 has been identified as the target of HBV X proteins and 17 out of 56 genes and TFs (p-value < 8.51e−4) have been identified as DNB members (Supplementary Table ‘KEGG enrichment analysis’). The findings are also highly related to the response of HBV infection in vivo, which provides more evidence of the vital functions of the DNB for the disease and further demonstrates the effectiveness of our method. The DNB of HBV sheds light on the progression of pathogen effects and also dynamically characterizes the disease progression in the liver.
To make a model-free prediction of the critical transition of diseases using a small number of samples of high-throughput data, we proposed a composite index serving as a leading early-warning indicator based on the DNB. Different from the previous works, our method for predicting complex diseases is based on searching for a particularly interacted network or a DNB carrying reliable information about dynamical variations. From the viewpoint of nonlinear dynamics, this network is driven by certain factors or parameters to first move to (or reflect the transition to) the disease state. Therefore, the network can serve as a new type of biomarker to detect the pre-disease state in a dynamical manner (Figs. 1f and h), in contrast to the traditional gene- or protein-based biomarkers that evaluate the system in a rather static manner.
On the other hand, because of disease complexities and personal variations (e.g., genetic or epigenetic factors), each individual may progress to the same disease through different underlying networks, which hampers the discovery of effective molecule- or model-based biomarkers. To overcome this difficulty, we have developed this model-free method based on measured individual data and therefore we can theoretically detect specific signals for each individual or potentially apply this method to personalized medicine. It should be noted that our prediction merely requires a few samples in each sampling period, in stark contrast to the consecutive time-series data over the entire period required by the traditional methods. Such early-warning signals can be applied to efficient prognosis as well as products of nichetargeting medicine. In addition to the theoretical conditions, we have also provided a computational algorithm to numerically identify the DNB based on high-throughput data. As the first step, this theoretical work is to challenge the complicated problem of detecting the pre-disease state by studying tissue based data. However, analyzing peripheral blood or plasma-based data is a practical approach to diagnosis at the molecular level, which we will consider in our future work. In addition, because different driving factors may result in different DNBs even for the same disease, the analysis of the difference and similarity of these DNBs is also an important future topic.
Derivation of DNB
The theoretical results were derived by considering the linearized equations17,27 of Eq.(1) with the perturbations of noise near . Namely, introducing new variables Y(k) = (y1(k), …, yn(k)) and a transformation matrix S, i.e., (Supplementary Information A1), we have
where Λ(P) is the diagonalized matrix of . ξ(k) = (ξ1(k), …, ξn(k)) are Gaussian noises with zero means and covariances κij = Cov(ξi, ξj). Actually, there are typically three cases arising in the diagonalization process and we present the detailed derivation in Supplementary Information A2 and here only illustrate the diagonal case with real eigenvalues for simplicity, i.e., Λ(P) = diag(λ1(P), …, λn(P)) with each |λi| between 0 and 1. Among the eigenvalues of Λ, the largest one in modulus, say λ1, first approaches to 1 when parameter P → Pc. This eigenvalue characterizes the system's rate of change around the fixed point and is called the dominant eigenvalue. Under the assumption that Y(k) have zero means, the covariance and PCC are respectively represented as
Note that the variance Var(yi) = Cov(yi, yi). Hence when the dominant eigenvalue |λ1| → 1 because of the change in the parameter values, Var(y1) → +∞, but Var(yi) is bounded for i ≠ 1 because 0 < |λi| < |λ1| < 1 (i = 2, …, n). For any i ≠ j, Cov(yi, yj) tends to have a positive bounded value.
Returning to the original variables Z whose values are directly measured as high-throughput data, by where sij is the element of the linear transformation S, we have (Supplementary Information A3)
Notice that variable y1 is related to the dominant eigenvalue λ1. From the above equations, it is obvious that as |λ1| → 1, the SD, i.e., increases greatly, or Var(zi) → + ∞ if si1 is not vanishing and |PCC(zi, zj)| approaches 1 drastically if both si1 and sj1 are non-zero. In this case, variables zi and zj are directly affected by the dominant eigenvalue. A group composed of such variables is called the dominant group in the network. On the other hand, as |λ1| → 1, |PCC(zi, zj)| between the dominant group (e.g., including zi) and others (e.g., including zj, which does not belong to the dominant group) reduces to zero if si1 ≠ 0 but sj1 = 0. Hence, we theoretically derived the three criteria to detect the dominant group or the DNB (see Supplementary Information A for more details). Clearly, a DNB can be viewed as a temporally dynamical module (TDM), whose members are strongly correlated with each other (i.e., a module feature) but their expressions stochastically fluctuate without keeping constant values (i.e., a dynamical or stochastic feature) only during the critical transition period (i.e., a temporal or transient feature). Hence, the presence of the DNB can be observed by requiring the measurement of not only correlations but also dynamics. In particular, it appears in the critical transition period but disappears (or merges with other components) in other periods, thereby signaling the upcoming critical transition. In addition to the disease progression, our theoretical results based on the bifurcation and center manifold theory23 may be widely applied to the detection of suddenly changing phenomena of various complex networks in a similar manner.
Genes in the DNBs for the three diseases have been linked and correlated by the combined functional couplings among them from various databases of protein-protein interactions of STRING, FunCoup and BioGrid, transcriptional regulations of TRED and metabolic pathways of KEGG (Supplementary Information F). Here, we considered the diseases of lung injury and HBV infection as examples to show the detailed computing procedure. We also compared the differences between our DNB-based method and the traditional methods, i.e., the fold-change method and the random gene selection method, based on respective gene sets (Supplementary Information E and G). Moreover, we also performed the pathway enrichment analysis of the identified DNBs (Supplementary Information F). The applicable computer program used for detecting the DNB from high-throughput data is described in the Supplementary Information of this paper.
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We thank Drs. Jiarui Wu and Rong Zeng of Chinese Academy of Sciences (CAS) for helpful discussions and also thank Drs. Masao Honda and Shuichi Kaneko of Kanazawa University for providing the data of reference 32. This work was supported by NSFC under No. 91029301, No. 61134013, No. 61072149 and No. 31100949, Shanghai Pujiang Program, the Chief Scientist Program of SIBS of CAS with No. 2009CSP002, the Knowledge Innovation Program of SIBS of CAS with Grant No. 2011KIP203, the Knowledge Innovation Program of CAS with No. KSCX2-EW-R-01, National Center for Mathematics and Interdisciplinary Sciences of CAS and JSPS/CSTP through the FIRST Program.
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Chen, L., Liu, R., Liu, ZP. et al. Detecting early-warning signals for sudden deterioration of complex diseases by dynamical network biomarkers. Sci Rep 2, 342 (2012). https://doi.org/10.1038/srep00342
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